The beginning of section 12, Calculus of variations (chapter 3, Variational principles) in V. I. Arnold's Mathematical Methods of Classical Mechanics (2nd edition, p. 55) reads:
For what follows, we will need some facts from the calculus of variations. A more detailed exposition can be found in (...), or G. E. Shilov, "Elementary Functional Analysis," MIT Press, 1974.
However, Shilov's book (at least the 1996 Dover print) doesn't seem to be about the calculus of variations at all. Does anyone know what's going on here?
Yes, the reference is not correct.
The correct reference (which can be found in the original Russian edition) is to Shilov's Mathematical Analysis: Special Course.